Theorem sbcbi2 3079

Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Assertion

Ref	Expression
sbcbi2	|- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Proof of Theorem sbcbi2

Step	Hyp	Ref	Expression
1		abbi 2343	. . 3 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2		eleq2 2293	. . 3 |- ( { x | ph } = { x | ps } -> ( A e. { x | ph } <-> A e. { x | ps } ) )
3	1, 2	sylbi 121	. 2 |- ( A. x ( ph <-> ps ) -> ( A e. { x | ph } <-> A e. { x | ps } ) )
4		df-sbc 3029	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
5		df-sbc 3029	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6	3, 4, 5	3bitr4g 223	1 |- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   [.wsbc 3028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  csbeq2  3148